Layer formation in double-diffusive convection over resting and moving heated plates

Abstract

We present a numerical study of double-diffusive convection characterized by a stratification unstable to thermal convection, while at the same time a mean molecular weight (or solute concentration) difference between top and bottom counteracts this instability. Convective zones can form in this case either by the stratification being locally unstable to the combined action of both temperature and solute gradients or by another process, the oscillatory double-diffusive convective instability, which is triggered by the faster molecular diffusivity of heat in comparison with that one of the solute. We discuss successive layer formation for this problem in the case of an instantaneously heated bottom (plate) which forms a first layer with an interface that becomes temporarily unstable and triggers the formation of further, secondary layers. We consider both the case of a Prandtl number typical for water (oceanographic scenario) and of a low Prandtl number (giant planet scenario). We discuss the impact of a Couette like shear on the flow and in particular on layer formation for different shear rates. Additional layers form due to the oscillatory double-diffusive convective instability, as is observed for some cases. We also test the physical model underlying our numerical experiments by recovering experimental results of layer formation obtained in laboratory setups.

Model validation

The numerical description and the physical model (Boussinesq approximation) are now validated against experimental and theoretical results. This way we show that our 2D simulation setup used in Sects. 2 and 3 provides a detailed physical description of layer formation in double-diffusive convection over heated bottom boundaries, apart from the differences that have to be accounted for to exactly reproduce the laboratory system. Since we are interested in a proof of concept, we have used a different simulation code which was easier to adapt for the laboratory comparison but much less suitable for the study we present in Sects. 2 and 3 (dimensionless units, need for excellent scaling in parallel computing, high approximation order, etc.).

First experiments of double-diffusive layering over a heated plate with a stable salinity gradient have been performed by Turner and Stommel [53]. The experimental setup in Turner and Stommel [53] consists of a tank with dimensions of ${0.25^3\,\hbox {m}^3}$. The initially stable salt water gradient measured 1% in density. The tank was heated from below at a heating rate of ${2\,\hbox {cals}\, \hbox {cm}^{-2}\,{\min }^{-1}}$, which gives ${1395.57\,\hbox {W\,m}^{-2}}$ in Si units, respectively. After 10 min the first layer formed at ${H=0.1\,\mathrm{m}}$. The final height of this first layer was reached after an hour at ${H=0.125\,\hbox {m}}$, whereas a second layer also formed above the first one during this hour. After ${95\,{\min }}$ three layers are visible, where the height of the second and the third are about ${0.037\,\hbox {m}}$ (see Fig. 2 in [53], the heating rate at the bottom was about ${1400\,\hbox {W\,m}^{-2}}$, the density gradient due to salinity measured 1%). The same experiment was repeated with doubled heating rates and steeper salinity gradients. Doubling the density gradient increased the amount of layers (Fig. 3a in [53], ${1400\,\hbox {W\,m}^{-2}}$ heat rate and a salinity gradient of 2%), whereas the doubled heating rate at the bottom increased the heights of the layers (Fig. 3b in [53], ${2800\,\hbox {W\,m}^{-2}}$ heating rate and a salinity gradient of 1%). However, the reader needs to keep in mind that the evolution of the layer formation is time-dependent. This makes it necessary to have time stamps available. In summary, only the first experiment can be used to validate a numerical code. Here, four time stamps and all physical properties are available.

The numerical setup follows the laboratory experiment where heat is imposed at the bottom (denoted with ‘bot’) by the heat flux $\mathbf{F}=k \nabla T$, which yields a Neumann boundary condition on temperature T,

$$\begin{aligned} \left( \frac{\partial T}{\partial y}\right) _{\mathrm{bot}} = F_{{y}}/k, \end{aligned}$$

(11)

with $F_{y}$ being the vertical component of $\mathbf{F}$ and k the thermal conductivity. The salinity gradient $\Delta S$ is defined by the density difference $\Delta \rho$ of 1% between the bottom and the top of the tank. The salinity is calculated by $\Delta S=\Delta \rho / (\rho \beta )$, cf. [52], Eqs. 7 and 18. All fluid properties are assumed to be constant at reference temperature $T_{\mathrm{ref}}=293\,\mathrm{K}$, cf. Table 1.

Table 1 Parameters used for the validation simulation

The top of the tank (denoted with ‘top’) is closed against the atmosphere fulfilling an insulator boundary condition. In the following, $(\frac{\partial T}{\partial y})_{\mathrm{top}} = 0$. This is justified by the following two premises: a) the radiative heat flux at the top $F_{\mathrm{top}}$ is three orders of magnitudes lower than $F_{\mathrm{bot}}$. b) the layering occurs only in the lower 2 / 3 of the tank, where the thermal influence of the upper boundary is negligible on the short time scales of the experiment. The side walls are insulated, too and fulfill $(\frac{\partial T}{\partial \mathbf{n}})_{{x}}= 0$. The solute boundaries are assumed to be impermeable for the solute at all boundaries, whence $(\frac{\partial S}{\partial \mathbf{n}})=0$. Impenetrable, stress-free boundary conditions are considered for the velocity field.

The numerical solution of the governing equations are calculated with the finite volume-based software OpenFOAM 5.x, [56]. More precisely, the set of equations given by Eq. (2) and the Boussinesq approximation Eq. (1) are solved with the PISO algorithm. Here, the momentum equation is solved by neglecting the pressure term. This results in a velocity field unfulfilling the incompressibility constraint. Subsequently, a Poisson equation is formulated to estimate the pressure. This pressure is used to correct the momentum field to fulfill $\nabla \cdot \mathbf{u}= 0$. This is the same numerical approach as used in ANTARES; however, the equations are treated in physical units. The entire simulation time of 5700 sec is identical with the last experimental snap shot.

The influence of the resolution is tested with a grid study. More precisely, rectangular grids with $200 \times 200$, $400 \times 400$, $800 \times 800$, and $1600 \times 1600$ are compared using parameters of the standard case. The main testing parameter is the height of the first layer, which is plotted for all four cases in Fig. 17. Only a small difference in the layer height of approx. 3% is found in comparison with selected 3D simulations. As the overall layer formation process is not influenced by the third spatial dimension, all simulations are conducted in 2D. The highest resolution resolves the saline boundary layer with two points and the thermal boundary layer with 20 points. This estimate is based on Eqs. (5) and (6) (see also [60], Appendix B.1). Obviously, the layer height decreases with higher resolutions. Additionally, the diffusive interface shows differences. In case of $200 \times 200$ and $400 \times 400$ cells the interface is very flat and the shape is not much influenced by the convective zone. This changes for $800 \times 800$ cells, where the interface gets turbulent and a unique layer height is not given any more. The interface shows structural changes in case of $1600 \times 1600$ cells. On top of the actual interface, a small mixing region is established. This additional layer is separated from the underlying zone, but important for the evolution of the second layer. It is neither purely diffusive nor purely convective, but stable enough not to be mixed with other layers. However, this connection layer is only visible at highest resolutions. In case of $800 \times 800$ the connection layer is merged into the first layer due to higher numerical diffusivity. The height of the first layer changes only by $<5\%$ for the highest resolutions. The grid study on the height of the first layer reveals a minimum resolution of $800 \times 800$ cells for simulating this experiment with OpenFoam. This is two times higher than our simulations performed with ANTARES. We suggest this to be a consequence of the second-order scheme used in OpenFoam.

Fig. 17

Height of the first layer as function of time for various resolutions. Theoretical and experimental results are depicted as straight lines and single full dots, respectively

In Turner [52] the authors described the growth of the first layer with an equilibrium model. They estimated the height by $h=C\cdot t^{1/2}S_*^{-1/2}H_*^{1/2}$, where $S_*=- g \beta (\frac{\mathrm{d}\,S}{\mathrm{d}\,y})$, $H_*=-\frac{g\alpha H}{\rho c}$ and $c=\sqrt{2}$. For known $H_*$ and $S_*$ the growth rate can be easily obtained and used to validate numerical models. Fernando [15] extended the range of the constant value c to $1.06<C<1.63$. The height of the first layer for both bounding values for parameters as given in Table 1 is depicted in Fig. 17. The height varies between 0.11–0.15 m after $t=6000\,\mathrm{s}$. Besides theoretical models we considered an experiment (cf. Fig. 2 in [53]), too. The layer heights are measured by hand and depicted as full dots in Fig. 17. However, only four snapshots are available in the original work. Further experiments are available from other authors, but the lack of experimental parameters makes it difficult to recreate them by numerical simulations. The height of layers in the experiment of Turner and Stommel [53] are as follows: 0.041 m after 600 s, 0.093 m after 1500 s, 0.112 m after 3600 s, and 0.142 m after 5700 s. Three of four values are within the plane spanned by the theoretical estimates (blue-yellow) for the first layer height. All tested resolutions are within this plane, too. A fitting curve of the four experimental points ranges within the two lower resolutions $200 \times 200$ and $400 \times 400$. This curve has an exponential growth rate of 0.5 and a constant $c=1.34$ for given values of $S_*$ and $H_*$. However, fluctuations of the layer height are assumed to be within 10%, which is based on uncertainties of the fluid properties and the thermal properties of the tank material. The numerical simulations act on the same time scales as the experiment. This means that the same amount of secondary layers are visible at comparable time stamps. Based on the comparison between theoretical estimates for the first layer height ([52] and [15]) and four experimental snap shots published in [53], we conclude that the governing equations given by Eqs. (1) and (2) are valid for simulations in the context of layer formation in double-diffusive convection.